SME
Grade 10 Math · Problem type
Area of a regular polygon
Use this for “A regular hexagon has sides of 4 cm. Find its apothem and its area.”
Diagram
A regular hexagon: 6 triangles meet at the center, each with base s and height a, the apothem
Half the center angle: θ = 180° ÷ 6 = 30°
K = ½ × 3.4641 × 24 = 41.5692 cm²
This is an example. Type your own values to start a new problem.
n
Number of sides
example
s
Side
example
P
Perimeter
calculated
θ
Half the angle at the center
worked out (not typed) · °
a
Apothem
worked out (not typed)
K
Area
worked out (not typed)
Formulas
P
=
n
×
s
24 = 6 × 4
θ
=
180
n
30
=
180
6
a
=
s
2 × tan(
θ
°)
3.4641
=
4
2 × tan(30°)
K
=
a
×
P
2
41.5692
=
3.4641
×
24
2
Assumptions
•
A regular polygon has n equal sides; the apothem a runs from the center to a side’s midpoint, at right angles.
•
Lines to the corners cut it into n triangles of base s and height a, so K = ½ × a × P.
•
Each triangle’s angle at the center is 360° ÷ n; the apothem halves it: θ = 180° ÷ n.
Step-by-step
WE KNOW
n = 6, s = 4 cm
FIND
perimeter (P), half the angle at the center (θ), apothem (a), area (K)
Step 1 · Find perimeter (P)
Use P = n × s
The
perimeter
is
n
equal
sides
of
length
s
.
P
=
6
×
4
P
=
24
cm
Step 2 · Find half the angle at the center (θ)
Use θ = 180 ÷ n
The
n
angles
at
the
center
make
360°,
and
the
apothem
cuts
each
in
half.
θ
=
180
6
θ
=
30°
Step 3 · Find apothem (a)
Use a = s ÷ (2 × tan(θ°))
In
the
right
triangle
at
the
center,
tan
θ
is
half
a
side
over
the
apothem.
a
=
4
2 × tan(30°)
a
=
4
2 × 0.5774
a
=
4
1.1547
a
=
3.4641
cm
Step 4 · Find area (K)
Use K = a × P ÷ 2
The
n
triangles
together:
1
2
×
apothem
×
all
their
bases.
K
=
3.4641
×
24
2
K
=
83.1384
2
K
=
41.5692
cm²
Check
Put the numbers back in. Both sides should match.
24 = 6 × 4 ✓
30
=
180
6
✓
3.4641
≈
4
2 × tan(30°)
✓
41.5692
=
3.4641
×
24
2
✓
Topic check
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